Explain why the number of solutions $\mathbf{x} \in \mathbb{R}^{3}$ of the simultaneous linear equations $A \mathbf{x}=\mathbf{b}$ is 0,1 or infinite, where $A$ is a real $3 \times 3$ matrix and $\mathbf{b} \in \mathbb{R}^{3}$. Let $\alpha$ be the mapping which $A$ represents. State necessary and sufficient conditions on $\mathbf{b}$ and $\alpha$ for each of these possibilities to hold.

Let $A$ and $B$ be $3 \times 3$ matrices representing linear mappings $\alpha$ and $\beta$. Give necessary and sufficient conditions on $\alpha$ and $\beta$ for the existence of a $3 \times 3$ matrix $X$ with $A X=B$. When is $X$ unique?

Find $X$ when

$$A=\left(\begin{array}{lll} 4 & 1 & 1 \\ 1 & 2 & 1 \\ 0 & 3 & 1 \end{array}\right), \quad B=\left(\begin{array}{lll} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 3 & 1 & 2 \end{array}\right)$$